Chapter 1. Characterisation of solid particlesWhat is particle technology?

Techniques for processing and handling particulate solids. It plays a major role in the production of materials in industry.

Chapter 1. Characterisation of solid particles

Individual solid particles are characterised by their size, shape, and density.

1.1 Particle shape

The shape of an individual particle is expressed in terms of the sphericity F s, which is independent of particle size. The sphericity of a particle is the ratio of the surface-volume ratio of a sphere with equal

volume as the particle and the surface-volume ratio of the particle. For a spherical particle of diameter D p, F s =1; for a non-spherical particle,

the sphericity is defined as

Dp: equivalent diameter of particle

Sp: surface area of one particle

vp: volume of one particle

The equivalent diameter is sometimes defined as the diameter of a sphere of equal volume. For fine particles, Dp is usually taken to be the

nominal size based on screen analysis or microscopic analysis. The surface area is found from adsorption measurements or from the

pressure drop in a bed of particles. For many crushed materials, F s is between 0.6 and 0.8. For particles rounded by abrasion, F smay be as

high as 0.95.

1.2 Particle size

In general "diameter" may be specified for any equidimensional particles. Particles that are not equidimensional, i.e. that are longer in

one direction than in others, are often characterised by the second

longest major dimension. For needle like particles, Dp would refer to the thickness of the particle, not their length. Units used for particle size

depend on the size of particles.

Coarse particles: inches or millimetres

Fine particles: screen size

Very fine particles: micrometers or nanometers

Ultra fine particles: surface area per unit mass, m2/g

1.3 Mixed particle sizes and size analysis

In a sample of uniform particles of diameter Dp, the total volume of the particles is m/r p, where m = mass of the sample, r p = density. Since the volume of one particle is vp, the total number of particle in the sample is

The total surface area of the particles is

To apply the above two equations to mixtures of particles having various size and densities, the mixture is sorted into fractions, each of constant

density and approximately constant size.

1.4 Specific surface of mixture

If the particle density r p and spericity F s are known, the surface area of particles in each fraction can be calculated and added to give the

specific surface, Aw.

where xi = mass fraction in a given increment,  = average diameter,

taken as arithmetic average of the smallest and largest particle diameters in increment.

1.5 Average particle size

(1). Volume-surface mean diameter,  , defined by

If the number of particles in each fraction Ni is known, then

(2). Arithmetic mean diameter

NT = number of particles in the entire sample

(3). Mass mean diameter

(4). Volume mean diameter

1.6 Number of particles in mixture

The volume of any particle is proportional to its "diameter" cubed.

a = volume shape factor

Assuming that a is independent of size

1.7 Screen analysis

Standard screens are used to measure the size (and size distribution) of particles in the size range between about 3 and 0.0015in (76mm and

38m m).

Screen is identified by meshes per inch, e.g. 10mesh, Dp = 1/10 = 0.1in.

The area of the openings in any one screen in the series is exactly twice that of the openings in the next smaller screen. The ratio of the actual

mesh dimension of any screen to that of the next smaller screen is =1.41.

Analysis using standard screen: Screens are arranged serially in a stack, with the smallest mesh at the bottom and the largest at the top.

Materials are loaded at top and then shacked for a period of time (e.g. 20 minutes).

14/20: through 14 mesh and on 20 mesh

Screen analysis gives: xi and  .Chapter 2. Motion of Particles through Fluids

2.1 Motion of particles through fluids

2.1.1 Mechanics of particle motion

Three forces acting on a particle moving through a fluid:

1). The external force, gravitational or centrifugal;

2). The buoyant force, which acts parallel with the external force but in the opposite direction;

3). The drag force, which appears whenever there is relative motion between the particle and the fluid

Drag: the force in the direction of flow exerted by the fluid on the solid is called drag.

2.1.2 Equations for one-dimensional motion of particle through fluid

Consider a particle of mass m moving through a fluid under the action of an external force Fe. Let the velocity of the particle relative to the fluid be u, let the buoyant force on the particle be Fb and let the drag

be FD, then

 (1)

The external force can be expressed as a product of the mass and the acceleration ae of the particle from this force,

 (2)

The buoyant force is, be Archimedes’ law, the product of the mass of the fluid displaced by the particle and the acceleration from the external

force. The volume of the particle is m/r p, the mass of fluid displaced is (m/r p)r , where r is the density of the fluid. The buoyant force is then

Fb = mr ae/r p (3)

The drag force is

FD = CDu2r Ap/2 (4)

where CD is the drag coefficient, Ap is the projected area of the particle

in the plane perpendicular to the flow direction.

By substituting the forces into Eq(1), we have

 (5)

Motion from gravitational force:

In this case, ae = g

 (6)

Motion in a centrifugal field:

ae = rw 2

 (7)

In this equation, u is the velocity of the particle relative to the fluid and is directed outwardly along a radius.

2.2 Terminal velocity

In gravitational settling, g is constant. Also, the drag always increases with velocity. The acceleration decreases with time and approaches zero. The particle quickly reaches a constant velocity which is the maximum attainable under the circumstances. This maximum settling velocity is

called terminal velocity.

 (8)

 (9)

In motion from a centrifugal force, the velocity depends on the radius and the acceleration is not constant if the particle is in motion with

respect to the fluid. In many practical use of centrifugal force, du/dt is small. If du/dt is neglected, then

 (10)

Motion of spherical particles:

If the particles are spheres of diameter Dp, then

m = p Dp3r p/6

Ap = p Dp2/4

Substitution of m and Ap into the equation for ut gives the equation for gravity settling of spheres:

 (11)

2.3 Drag coefficient

Drag coefficient is a function of Reynolds number. The drag curve applies only under restricted conditions:

i). The particle must be a solid sphere;

ii). The particle must be far from other particles and the vessel wall so that the flow pattern around the particle is not distorted;

iii). It must be moving at its terminal velocity with respect to the fluid.

Particle Reynolds number:

 (12)

u: velocity of approaching stream

Dp: diameter of the particle

r : density of fluid

m : viscosity of fluid

Stokes’ law applies for particle Reynolds number less than 1.0

CD = 24/NRe,p (13)

From Eq(4)

FD = 3p m ut Dp (14)

From Eq(11)

ut = g Dp2(r p - r )/(18m ) (15)

At NRe,p =1, CD =26.5 instead of 24 from the above equation.

Centrifugal: rw 2 ® g.

For 1000 < NRe,p <200,000, use Newton’s law

CD = 0.44 (16)

FD= 0.055p Dp2 ut2r (17)

 (18)

Newton’s law applies to fairly large particles falling in gases or low viscosity fluids.

Terminal velocity can be found by trial and error after guessing NRe,p to get an initial estimate of CD.

2.4 Criterion for settling regime

To identify the range in which the motion of the particle lies, the velocity term is eliminated from the Reynolds number by substituting utfrom

Stokes’ law

 (19)

If Stokes’ law is to apply, NRe,p <1.0. Let us introduce a convenient criterion K

 (20)

Then NRe,p = K3/18. Setting NRe,p = 1 and solving for K gives K=2.6. If K is less than 2.6 then Stokes’ law applies.

Substitution for ut using Newton’s law

NRe,p = 1.75K1.5

Setting NRe,p = 1000 and solving for K gives K = 68.9. Setting NRe,p = 200,000 and solving for K gives K = 2,360.

· Stokes’ law range: K < 2.6

· Newton’s law range: 68.9 < K < 2,360

· when K > 2,360 or 2.6 < K < 68.9, ut is found from   using a value of CD found by trial from the curve.

2.5 Hindered settling

In hindered settling, the velocity gradients around each particle are affected by the presence of nearby particles. So the normal drag

correlations do not apply. Also, the particles in settling displace liquid, which flows upward and make the particle velocity relative to the fluid greater than the absolute settling velocity. For uniform suspension, the settling velocity us can be estimated from the terminal velocity for an

isolated particle using the empirical equation of Maude and Whitmore

us = ut(e )n

Exponent n changes from about 4.6 in the Stokes’ law range to about 2.5 in the Newton’s law region. For very small particles, the calculated

ratio us/ut is 0.62 for e =0.9 and 0.095 for e =0.6. With large particles, the corresponding ratios are us/ut = 0.77 and 0.28; the hindered settling

effect is not as profound because the boundary layer thickness is a smaller fraction of the particle size.

If particles of a given size are falling through a suspension of much finer solids, the terminal velocity of the larger particles should be calculated

using the density and viscosity of the fine suspension. The Maude-Whitmore equation may then be used to estimate the settling velocity

with e taken as the volume fraction of the fine suspension, not the total void fraction.

Suspensions of very fine sand in water is used in separating coal from heavy minerals and the density of the suspension is adjusted to a value slightly greater than that of coal to make the coal particles rise to the

surface, while the mineral particles sink to the bottom.Chapter 3. Size Reduction

Four commonly used methods for size reduction: 1). Compression; 2). Impact; 3). Attrition; 4). Cutting.

3.1 Principle of size reduction

Criteria for size reduction

An ideal crusher would (1) have a large capacity; (2) require a small power input per unit of product; and (3) yield a product of the single size distribution desired.

Energy and power requirements in size reduction

The cost of power is a major expense in crushing and grinding, so the factors that control this cost are important.

3.2 Crushing efficiency

3.2.1 Empirical relationships: Rittinger’s and Kick’s law

The work required in crushing is proportional to the new surface created. This is equivalent to the statement that the crushing efficiency is constant and, for a giving machine and material, is independent of the sizes of feed and product. If the sphericities  a (before size reduction) and  b (after size reduction) are equal and the machine efficiency is constant, the Rittinger’s law can be written as

where P is the power required,   is the feed rate to crusher,   is the average

particle diameter before crushing,   is the average particle diameter after crushing, and Kr is Rittinger’s coefficient.

Kick’s law: the work required for crushing a given mass of material is constant for the same reduction ratio, that is the ratio of the initial particle size to the finial particle size

where Kk is Kick’s coefficient.

3.2.2 Bond crushing law and work index

The work required to form particles of size Dp from very large feed is proportional to the square root of the surface-to-volume ratio of the product, sp/vp. Since  s = 6/Dp, it follows that

where Kb is a constant that depends on the type of machine and on the material being crushed.

The work index, wi, is defined as the gross energy required in KWH per ton of feed to reduce a very large feed to such a size that 80% of the product passes a 100 m screen. If Dp is in millimetres, P in KW, and   in tons per hour, then

If 80% of the feed passes a mesh size of Dpa millimetres and 80% of the product a mesh of Dpb millimetres, it follows that

Example: What is the power required to crush 100 ton/h of limestone if 80% of the feed pass a 2-in screen and 80% of the product a 1/8 in screen? The work index for limestone is 12.74.

Solution:  =100 ton/h, wi =12.74, Dpa =2 25.4=50.8 mm, Dpb =25.4/8=3.175 mm

3.3 Size reduction equipment

Size reduction equipment is divided into crushers, grinders, ultrafine grinders, and cutting machines. Crusher do the heavy work of breaking large pieces of solid material into small lumps. A primary crusher operates on run-of -mine material accepting anything that comes from mine face and breaking it into 150 to 250 mm lumps. A secondary crusher reduces these lumps into particles perhaps 6mm in size. Grinders reduce crushed feed to powder. The product from an intermediate grinder might pass a 40-mesh screen; most of the product from a fine grinder would pass a 200-mesh screen with a 74 m opening. An ultrafine grinder accepts feed particles no larger than 6mm and the product size is typically 1 to 5 m. Cutters give particles of definite size and shape, 2 to 10mm in length.

The principal types of size-reduction machines are as follows:

A. Crushers (coarse and fine)

1. Jaw crushers2. Gyratory crushers3. Crushing rolls

B. Grinders (intermediate and fine)

1. Hammer mills; impactors2. Rolling-compression mills3. Attrition mills4. Tumbling mills

C. Ultrafine grinders

1. Hammer mills with internal classification2. Fluid-energy mills3. Agitated mills

D. Cutting machines

1. Knife cutters; dicers; slitters

Chapter 4. Mechanical SeparationsMechanical separations are performed based on the physical difference between particles such as size, shape, or density. Mechanical separations are applicable to heterogeneous mixtures, not to homogeneous solutions.

4.1 Screening

Screening is a method of separating particles according to size alone.

Undersize: fines, pass through the screen openings

Oversize: tails, do not pass

A single screen can make but a single separation into two fractions. These are called unsized fractions, because although either the upper or lower limit of the particle sizes they contain is known, the other limit is unknown. Material passed through a series of screens of different sizes is separated into sized fractions, i.e. fractions in which both the maximum and minimum particle sizes are known.

4.1.1 Screening equipment

Stationary screens and grizzlies; Gyrating screens; Vibrating screens; Centrifugal sitter.

Cutting diameter Dpc: marks the point of separation, usually Dpc is chosen to be the mesh opening of the screen.

Actual screens do not give a perfect separation about the cutting diameter. The undersize can contain certain amount of material coarser than Dpc, and the oversize can contain certain amount of material that is smaller than Dpc.

4.1.2 Material balances over a screen

Let F, D, and B be the mass flow rates of feed, overflow, and underflow, respectively, and xF, xD, and xB be the mass fractions of material A in the streams. The mass fractions of material B in the feed, overflow, and underflow are 1- xF, 1- xD, and 1- xB.

F = D + B

FxF = DxD + BxB

Elimination of B from the above equations gives

Elimination of D gives

4.1.3 Screen effectiveness

A common measure of screen effectiveness is the ratio of oversize material A that is actually in the overflow to the amount of A entering with the feed. These quantities are DxD and FxF respectively. Thus

where EA is the screen effectiveness based on the oversize. Similarly, an effectiveness EB based on the undersize materials is given by

A combined overall effectiveness can be defined as the product of the two individual ratios.

Example: A quartz mixture is screened through a 10-mesh screen. The cumulative screen analysis of feed, overflow and underfolw are given in the table. Calculate the mass ratios of the overflow and underflow to feed and the overall effectiveness of the screen.

Mesh Dp (mm) Feed Overflow Underflow

4 4.699 0 0 0

6 3.327 0.025 0.071 0

8 2.362 0.15 0.43 0

10 1.651 0.47 0.85 0.195

14 1.168 0.73 0.97 0.58

20 0.833 0.885 0.99 0.83

28 0.589 0.94 1.0 0.91

35 0.417 0.96   0.94

65 0.208 0.98   0.975

Pan   1.0   1.0

Solution:

From the table, xF=0.47, xD=0.85, xB=0.195

4.1.4 Capacity and effectiveness of screens

The capacity of a screen is measured by the mass of material that can be fed per unit time to a unit area of the screen. Capacity and effectiveness are opposing factors. To obtain maximum effectiveness, the capacity must be small, and large capacity is obtainable only at the expense of a reduction in effectiveness.

4.2 Filtration

Filtration is the removal of solid particles from a fluid by passing the fluid through a filtering medium, or septum, on which the solids are deposited. The fluid may be liquid or gas, the valuable stream from the filter may be fluid, or the solid, or both. Sometimes it is neither, as when waste solid must be separated from waste liquid prior to disposal.

Filters are divided into three main groups: cake filters, clarifying filters, and crossflow filters. Cake filters separate relatively large amount of solids as a cake of crystals or sludge. Often they include provisions for washing the cake and for removing some of the liquid from the solids before discharge. At the start of filtration in a cake filter, some solid particles enter the pores of the medium and are immobilised, but soon others begin to collect on the septum surface. After this brief period the cake of solids does the filtration, not the septum; a visible cake of appreciable thickness builds up on the surface and must be periodically removed. Clarifying filters remove small amount of solids to produce a clean gas or a sparkling clear liquid such as beverage. The solid particles are trapped inside the filter medium or on its external surfaces. Clarifying filters differ from screens in that the pores of the filter medium are much larger in diameter than the particles to be removed. In a crossflow filter, the feed suspension flows under pressure at a fairly high velocity across the filter medium. A thin layer of solids may form on the surface of the medium, but the high liquid velocity keeps the layer from building up. The filter medium is a ceramic, metal, or polymer membrane with pores small enough to exclude most of suspended particles. Some of the liquid passes through the medium as clear filtrate, leaving a more concentrated suspension behind.

4.3 The theory of filtration

In cake filters, the particles forming the cake are small and the flow through the bed is slow. Streamline conditions are invariably obtained. From Kozeny equation,

 (1)

where u is the velocity of the filtrate, L is the cake thickness, S is the specific surface of the particles,  is the porosity of cake,  is the viscosity of the filtrate, and  P is the applied pressure difference. The filtrate velocity can also be written as

 (2)

where V is the volume of filtrate which has passed in time t and A is the total cross-sectional area of the filter cake.

For incompressible cakes  can be taken as constant and the quantity  3/[5(1- )2S2] is then a property of the particles forming the cake and should be constant for a given material. Therefore

 (3)

where

 (4)

Eq(3) is the basic filtration equation and r is termed the specific resistance. It is seen to depend on  and S. For incompressible cakes it is taken as constant, but it will depend on the rate of deposition, nature of particles, and on forces between the particles.

In Eq(3), the variables V and L are connected, and the relation between them can be obtained by making a material balance between the solids in the slurry and in the cake.

Mass in the filter cake is (1- )AL s, where  s is the density of the solids.

Mass of liquid retained in the filter cake is  AL , where  is the density of the filtrate.

If J is the mass fractions of solids in the original suspension

 (5)

That is

 (6)

Therefore

 (7)

and

 (8)

If v is the volume of cake deposited by unit volume of filtrate then:

 or   (9)

and from Eq(8):

 (10)

Substituting for L in Eq(3)

or

 (11)

Eq(11) can be regarded as the basic relation between  P, V, and t. Two important types of operation will be considered: 1). where the pressure difference is maintained constant and, 2). where the rate of filtration is maintained constant.

Constant pressure difference

Eq(11) can be re-written as

 (12)

Integrating Eq(12) gives

 or   (13)

Thus for a constant pressure filtration, there is a linear relation between V2 and t. Filtration at constant pressure is more frequently adopted in practical conditions.

Constant rate filtration

constant (14)

Therefore

 or   (15)

In this case,  P is directly proportional to V.

Flow of filtrate through the septum and cake combined

Suppose that the filter septum to be equivalent to a thickness Ls of cake, then if  P is the pressure drop across the cake and septum combined Eq(3) can be written as:

 (16)

i.e.

 (17)

For constant rate filtration we have

 (18)

For constant pressure filtration we have

 (19)

4.4 Separations based on the motion of particles through fluids

Devices that separate particles of differing densities are known as sorting classifiers. They use one of the two principal separation methods: sink-and-float and differential settling.

4.4.1. Sink-and-float methods

A sink-and-float method uses a liquid sorting medium, the density of which is intermediate between that of the light material and that of the heavy material. Then the heavy particles settle through the medium, and the lighter ones float, and a separation is thus obtained. This method has the advantage that, in principle, the separation depends only on the difference in the densities of the two substances and is independent of the particle size. This method is also known as the heavy-fluid separation.

Heavy fluid processes are used to treat relatively coarse particles, usually greater than 10-mesh. A comment choice of medium is a pseudoliquid consisting of a suspension in water of fine particles

4.4.2. Differential settling methods

Differential settling methods utilise the difference in terminal velocities that exist between substances of different density. The density of the medium is less than that of either substance.

Consider particles of two materials A and B settling through a medium of density  . Let A be the heavier. If the smallest particle of A settles faster than the largest particle

of B, then complete separation of A and B can be achieved.

For settling in the Stokes’ law region, the terminal velocity can be calculated as

For equal-settling particles, utA = utB, therefore

For settling in the Newton’s law range

If the ratio of diameters of the smallest particle of A and the largest particle of B is larger than the equal-settling ratio, then perfect separation of A and B can be achieved.